The course will give an introduction to a broad range of insulating, metallic and superconducting topological materials. We will see how topological phases do not appear only in electronic systems, but characterize more broadly all condensed matter. After a quick recap of basic two-dimensional and three-dimensional topological insulators, we will introduce topological crystalline insulators. Then, we will discuss the Chern-Simons form and axion coupling in the context the magnetoelectric effect, leading to axion insulators. Metals can also be topological: we will introduce Weyl fermions and related phenomena such as Fermi arcs and chiral anomaly. We will discuss topological superconductors in one and two dimensions that lead to Majorana modes at the boundary and vortex cores, with a quick outline of braiding and implications for topological quantum computing. Then, we will discuss topological order and the fractional quantum Hall effect. 

Note: This course assumes students are familiar with the topics covered in the basic course Electronic structure and quantum simulation of materials III which I will offer immediately before, in January-March 2025.

NB: Lecture time schedule is tentative.

References used in the course:

• D. Vanderbilt, Berry Phases in Electronic Structure Theory, Cambridge University Press
• B. A. Bernevig with T. Hughes, Topological Insulators and Topological Superconductors, Princeton University Press
• R. Resta’s Lecture Notes on Geometry and Topology in Electronic Structure Theory (http://www-dft.ts.infn.it/~resta/gtse/draft.pdf) 
• Scientific articles that will be cited in class